PaperlayerView exact revision on arXiv

Source-linked AI summary

Functional Autoencoders for Amplitude-Phase Representation Learning

Peida Wu, Xinyang Xiong, Pengcheng Zeng

arXiv:2609.34207v1stat.MLcs.LG

TL;DR

Functional data can combine nonlinear shape variation with temporal misalignment, which linear dimension reduction and functional autoencoders without explicit warping do not fully address. AP-FAE learns separate amplitude and phase embeddings with a shared warp under a reconstruction objective. It outperforms baselines on most clustering and alignment metrics and all reconstruction metrics across synthetic and six real-world benchmarks.

  • Problem

    Linear dimension reduction can restrict nonlinear amplitude variation, while functional autoencoders without explicit warping entangle temporal misalignment with shape.

    Source: §1 · pp. 1–2

  • Method

    AP-FAE jointly encodes all channels into separate amplitude and phase embeddings, reconstructs channel-specific canonical-time functions, and models phase with a shared monotone, endpoint-preserving warp.

    Source: §1 · p. 2

  • Results

    AP-FAE outperforms state-of-the-art baselines on most clustering and alignment metrics and all reconstruction metrics across synthetic and six real-world benchmarks.

    Source: §1 · p. 3

  • Takeaways & Limitations

    On real data, clustering with amplitude embeddings alone consistently surpasses clustering with joint amplitude–phase embeddings.

    Source: §1 · p. 3

    AP-FAE assumes a continuous, positive-velocity warp shared across channels and performs worse when true warps have long constant segments or upward jumps, or observations have severe missingness or strongly uneven sampling.

    Source: §7 · p. 11

Abstract

from arXiv · show

Functional data are intrinsically infinite-dimensional, and often exhibit phase variation, where corresponding events occur at different times across observations. Existing linear dimension reduction methods struggle with nonlinear amplitude variation, while functional autoencoders without an explicit warp entangle temporal misalignment with shape. We propose the Amplitude--Phase Functional Autoencoders (AP-FAE), an unsupervised framework for functional data that spans both univariate and multivariate cases, with emphasis on the multivariate setting, and factorizes the latent space into separate amplitude and phase embeddings derived from all channels. A smooth functional decoder reconstructs channel-specific amplitude functions in canonical time, and a shared monotone, endpoint-preserving warp captures phase variation. We prove a bound linking amplitude recovery to registration, reconstruction, and noise errors, and validate it numerically. Across synthetic data and six real-world benchmarks, AP-FAE outperforms state-of-the-art baselines on most clustering and alignment metrics and on all reconstruction metrics. Clustering with amplitude embeddings alone consistently surpasses joint amplitude--phase clustering, confirming the benefit of explicit disentanglement. Code is available at https://anonymous.4open.science/r/APFAE-418C/}{https://anonymous.4open.science/r/APFAE-418C/.

1 Introduction

Phase variation can make similar-shaped curves appear different, while linear methods and functional autoencoders without explicit warping have limits in representing nonlinear amplitude and timing. AP-FAE separates amplitude and phase in an unsupervised reconstruction framework, and the paper reports strong benchmark results and a bound on amplitude recovery.

1 Introduction

  • Phase variation shifts corresponding events across time, so curves with similar shapes can appear different and representations may reflect timing rather than shape.Traditional registration is separate from the learning objective, allowing alignment errors to propagate.Source: §1 · p. 1
  • Linear dimension reduction can restrict nonlinear variation, while functional autoencoders without explicit warping entangle temporal misalignment with amplitude.Channel-wise FPCA also ignores cross-channel covariance; multivariate FPCA remains linear.Source: §1 · p. 2
  • AP-FAE jointly encodes all channels into separate amplitude and phase embeddings, decoding canonical-time channel trajectories and a shared monotone, endpoint-preserving warp.It is trained by reconstruction alone and applies to univariate and multivariate data, with primary focus on multivariate settings.Source: §1 · p. 2
  • The paper establishes a bound linking amplitude recovery to registration, reconstruction, and noise errors, with numerical validation on synthetic data.The authors describe it as the first such bound for nonlinear functional representation learning.Source: §1 · p. 2
  • AP-FAE outperforms state-of-the-art baselines on most clustering and alignment metrics and all reconstruction metrics across synthetic and six real-world benchmarks.On real data, clustering with amplitude embeddings alone consistently surpasses clustering with joint amplitude–phase embeddings.Source: §1 · p. 3

2 Related Work

Prior work addresses functional alignment, linear dimension reduction, and smooth functional autoencoding, but these approaches leave gaps in task-independent, nonlinear amplitude–phase representation. AP-FAE combines separate embeddings with multivariate functional decoding under a reconstruction-only objective.

Functional alignment and amplitude–phase separation

  • Joint alignment methods integrate registration with clustering or classification, but their representations are tied to specific tasks; AP-FAE instead learns separate amplitude and phase embeddings under reconstruction alone.Clustering is applied to amplitude embeddings after fitting and does not influence alignment.Source: §2 · p. 3

Linear functional dimension reduction

  • FPCA and MFPCA provide linear dimension reduction, but may require many components for nonlinear or misaligned curves, and standard MFPCA lacks a subject-specific warp.Nonlinear FPCA extensions often rely on kernel or manifold assumptions and do not jointly model temporal deformation.Source: §2 · p. 3

Autoencoders and functional autoencoders

  • Functional autoencoders provide smooth reconstruction, but without explicit temporal deformation, temporal shifts mix with shape variation or consume latent capacity.AP-FAE adds a separate phase code and warping network to model temporal variation explicitly.Source: §2 · p. 3

3 The AP-FAE Framework

AP-FAE learns separate amplitude and phase embeddings for functional data, reconstructing channel-specific shapes in canonical time and modeling timing with a shared warp. The components are trained jointly through reconstruction, with the amplitude code intended to capture shape and the phase code where that shape occurs in time.

Overview

  • AP-FAE maps functional observations to separate amplitude and phase embeddings, then jointly trains decoders to reconstruct channel-specific amplitudes and a shared inverse warp.The framework is presented for univariate and multivariate data, with primary focus on the multivariate setting.Source: §3 · p. 4, Figure 2 · p. 4

Functional encoder

  • The encoder extracts features through functional inner products with fixed cubic B-spline weights before forming embeddings from information across all channels.For discrete observations, the inner products are evaluated by numerical quadrature on a common grid, followed by a ReLU and affine map.Source: §3 · p. 5

Amplitude decoder

  • The amplitude decoder maps the amplitude embedding to channel-specific trajectories on canonical time, using channel-specific coefficients and not the phase embedding.It uses the same fixed B-spline bases as the encoder, with a shared amplitude embedding across channels.Source: §3 · p. 5

Phase decoder and time warping

  • The phase network uses the phase code to produce a strictly increasing, endpoint-preserving inverse warp shared across channels.Positive warp density enforces monotonicity, while normalization fixes the endpoints at 0 and 1.Source: §3 · p. 5

Reconstruction and learning

  • Reconstruction evaluates each decoded amplitude at warped time, and joint training minimizes reconstruction error; the amplitude embeddings are used for clustering, while both codes are reserved for settings where phase is predictive.The paper assumes a common aligned coordinate and models the observed curves as amplitude functions composed with shared-across-channel inverse warps, plus measurement noise.Source: §3 · pp. 4, 6

4 Theory

The theory bounds amplitude-recovery error in terms of registration, reconstruction, and measurement-noise errors under stated regularity conditions. It also emphasizes that low reconstruction error alone does not ensure accurate amplitude recovery.

Amplitude error bound

  • The amplitude-recovery bound relates errors in the fitted amplitude curves to registration, reconstruction, and measurement noise.The result assumes Lipschitz true amplitude curves and strictly increasing continuous bijective true and fitted warps, with fitted warps also C-Lipschitz.Source: §4 · p. 6

Practical implication and feasibility

  • The bound measures reconstruction error against noisy observations and requires no independence or zero-mean assumption on the noise.The fitted warps are strictly increasing by construction, and bounded-slope piecewise-linear or smooth warps satisfy the stated Lipschitz condition.Source: §4 · pp. 6–7
  • Small reconstruction error alone does not guarantee accurate amplitude recovery; reliable amplitude embeddings also require accurate phase alignment.The discussion identifies reconstruction error, registration error, and noise level as separately measurable quantities.Source: §4 · p. 6
  • The bound is satisfied across the reported simulation scenarios, and amplitude and phase are identifiable only relative to a common aligned coordinate.The theorem conditions on registration accuracy in that coordinate rather than deriving it from reconstruction loss alone.Source: §4 · p. 7

5 Experiments

Across synthetic and six real-world benchmarks, AP-FAE performs strongly across clustering, alignment, and reconstruction, while experiments also assess recovery, embedding choices, and robustness of the evaluation design.

5 Experiments

  • The evaluation covers trivariate synthetic curves and six UEA/UCR datasets, using ACC, ATV, and MSE; comparisons are partial because baselines report different task combinations.Synthetic data vary cross-channel correlation ρ and phase-distortion range B; the real benchmarks include three multivariate and three univariate datasets.Source: §5 · p. 7, §5.1 · p. 7, §5.2 · p. 9

5.1 Simulation

  • AP-FAE leads ACC, ATV, and MSE in all four synthetic settings; at ρ = 0.2, B = 0.75, ACC is 0.924 versus 0.875 for SRC.FAE clustering accuracy falls as B increases, while AP-FAE remains stable; reconstruction MSE is compared only with FAE and FAEclust.Source: §5.1 · p. 7, Table 1 · p. 8
  • Estimated amplitude components and inverse warps closely agree with their simulation ground truths.Figure 3 compares the first two multivariate FPCs across channels 1–3 and inverse warps for six subjects.Source: §5.1 · p. 8, Figure 3 · p. 8
  • All 200 simulation fits satisfy the amplitude-error bound, with errors of 0.214–0.253 and bounds of 0.434–0.523, confirming the theory while showing it is conservative.Source: §5.1 · pp. 8–9

5.2 Real-World Data

  • AP-FAE achieves the best ACC on five of six real datasets, best ATV on five, and best MSE on all six, with SRC leading ACC on UWaveGestureLibrary and ATV on ERing.FAE and FAEclust are MSE runners-up on every dataset but never best in clustering; Figure 4 visualizes one UWaveGestureLibrary fit.Source: §5.2 · p. 9, Figure 4 · p. 9, Table 2 · p. 10
  • Clustering with amplitude embedding a alone performs best; adding phase embedding p degrades performance, while p alone can remain informative for tasks such as regression.Across the tested embedding-size pairs, r = q = 4 is consistently competitive and is used throughout.Source: §5.2 · p. 9, Table 3 · p. 10

6 Discussion

The discussion tests robustness to sampling and warp deviations and reports scalability across sample sizes and grid resolutions.

Embedding-size sensitivity

  • Figure 5 marks the best-performing embedding-size settings with stars.Source: Figure 5 · p. 10

Robustness to irregular sampling and missing observations

  • Clustering ACC stays near the complete-data baseline under missingness and uneven sampling, but sparse temporal coverage can impair reconstruction and alignment.With 80% random missingness, MSE and ATV are 0.0072 and 1.212; strongly uneven sampling raises them to 0.0302 and 1.275.Source: §6 · p. 10

Extreme warping or no warping

  • Plateaus and jumps increase ATV and MSE because AP-FAE assumes continuous, strictly positive-velocity warps and cannot represent those patterns exactly.With no warping, AP-FAE performs comparably to FAE and estimates the identity warp.Source: §6 · p. 11

Scalability

  • Training time grows approximately linearly with subject count, while a 33-fold increase in grid points changes training time by less than 7%.From 90 to 30,000 subjects, MSE falls from 0.016 to 0.011; normalized ATV stays within 0.116–0.118 across grid resolutions.Source: §6 · p. 11

7 Conclusion

AP-FAE separates amplitude and phase representations and uses a shared warp to model temporal variation; the paper reports benchmark gains and identifies settings where warp recovery is unreliable. The supplied material also describes the training algorithm, AI-use disclosure, and reproducibility resources.

7 Conclusion

  • Across synthetic and six real-world benchmarks, AP-FAE outperformed state-of-the-art baselines on most metrics, but its performance degrades for discontinuous or otherwise challenging warps and sampling.Stable clustering or low average reconstruction error in these settings should not be interpreted as recovery of the true warp.Source: §7 · p. 11
  • The framework assumes a continuous, positive-velocity warp shared across channels, and fixes the amplitude and phase embedding sizes rather than learning them from data.The authors identify data-adaptive embedding sizes and richer warp families as future work.Source: §7 · p. 11

AI use statement

  • The authors report using generative ChatGPT and Deepseek for manuscript polishing and code implementation, with all AI-assisted edits and code reviewed by the authors.They state that AI tools were not used for research ideation, method design, or analysis and interpretation of results.Source: AI use statement · p. 12

Reproducibility statement

  • The code, training and evaluation scripts, proofs, assumptions, experimental setup, baseline settings, and synthetic-data procedure are identified as reproducibility resources.The source code is available at the anonymous repository linked in the abstract.Source: Reproducibility statement · p. 12

A.1 Algorithms

  • The training loop encodes each observation into amplitude and phase embeddings, computes and normalizes phase-dependent warp densities, and updates the model using reconstruction loss.The algorithm also computes channel-specific amplitude-decoder outputs from the amplitude embeddings.Source: §A.1 · p. 15

A.2 Phase Decoder Implementation

The phase decoder constructs an inverse warp from positive density values and interpolates decoded amplitude trajectories at warped times. The appendix also describes computational scaling and the assumptions behind the amplitude-recovery bound.

A.2 Phase Decoder Implementation

  • Positive increments make the inverse warp strictly monotone, while normalization sets its endpoint to 1; subtracting the maximum stabilizes exponentiation without changing the normalized warp.Interpolation uses the same warp locations and weights across channels, with gradients passing through both amplitude values and warp locations.Source: §A.2 · p. 16

A.3 Computational and Space Complexity

  • With grid resolution, basis size, network widths, and minibatch size fixed, training cost scales linearly in subjects and channels, while the shared warp avoids an additional channel factor in the phase pathway.The stated per-epoch cost accounts for encoding, amplitude decoding, warping, interpolation, and reconstruction loss.Source: §A.3 · p. 16

B Details on the Theory

  • The amplitude-recovery bound is proved for each noise realization without distributional assumptions on the noise.The proof defines the reconstruction error against noisy observations and describes the reconstruction and noise levels as root mean squared L2 quantities across subjects.Source: B Details on the Theory · pp. 16–17

C.1 Evaluation Metrics

The evaluation passage names clustering accuracy and an alignment criterion, and notes that the latter uses class labels but not true warping functions. The baseline details list functional autoencoders, clustering methods, and alignment or registration approaches.

C.1 Evaluation Metrics

  • The evaluation names clustering accuracy (ACC) and the pairwise ATV criterion for alignment; ATV uses class labels but requires no true warping functions.The supplied text does not provide a complete definition of ACC or ATV.Source: §C.1 · pp. 17–18

C.2 Baseline Implementation Details

  • The listed non-registration baselines include FAE, FAEclust, JPCCA, Funclust, and funHDDC, with implementations drawn from reported settings or published implementations.FAE is described as a functional autoencoder without phase variation (Wu et al., 2024); FAEclust uses elastic distance (Singh et al., 2025).Source: §C.2 · p. 18
  • SrvfRegNet is locally reproduced with a shared-warp multichannel extension and provides alignment-only entries; SRC estimates a shared subject-specific warp without class labels.The SRC setup includes a scalar univariate specialization and a coordinate-list extension for trivariate curves.Source: §C.2 · p. 18

C.3 Synthetic Data Generation

The simulations validate the amplitude-recovery bound across four settings, using synthetic trivariate curves with class-specific amplitude patterns and subject-specific shared warps. The design varies cross-channel coefficient correlation and warp severity.

C.3 Synthetic Data Generation

  • The main synthetic comparison uses three classes of trivariate curves, with 100 subjects per class and 100 equally spaced observations on [0, 1].Amplitude curves are constructed from three harmonics with class-specific coefficient means.Source: §C.3 · pp. 18–19
  • Each subject has one increasing warp shared across channels that fixes both endpoints; warps are independent of amplitude coefficients and class membership.The simulation varies the distortion range B between 0.75 and 1.25 and cross-channel coefficient correlation ρ between 0.2 and 0.6.Source: §C.3 · p. 19

C.4 Numerical Validation of the Amplitude Recovery Bound

  • All 200 fits satisfied the amplitude-recovery bound; mean amplitude errors were 0.214–0.253 versus mean bounds of 0.434–0.523.The table reports means over 50 fits per simulation setting, and the authors describe the bound as conservative in these simulations.Source: §C.4 · p. 19, Table A1 · p. 20

C.5 Details of the Robustness Experiments

The robustness experiments test AP-FAE under missing or uneven observations and extreme warps, using repeated synthetic fits; a separate identity-warp simulation compares AP-FAE with FAE and FAEclust.

C.5 Details of the Robustness Experiments

  • Robustness tests use three channels, three classes, 100 subjects per class, 50 repetitions, and retrain AP-FAE for each condition.The missingness experiment uses B = 0.75 and ρ = 0.2 and evaluates ACC, ATV, and MSE against a dense noise-free reference.Source: §C.5 · p. 20, Table A2 · p. 20
  • Sampling tests remove 20%, 50%, or 80% of coordinates while retaining endpoints, or concentrate observations near the domain end at two strengths.Uneven sampling selects interior points from a 1001-point grid; all subjects and channels share each replication’s grid.Source: §C.5 · p. 21
  • Plateau-warp tests compare one pause of length 0.04, 0.08, or 0.10 with two separated pauses totaling 0.08.These continuous, endpoint-preserving warps test the strict positive-velocity assumption because the true warp has zero derivative on a nonempty interval.Source: §C.5 · pp. 20–21, Table A3 · p. 20
  • Jump-warp tests use nondecreasing, endpoint-preserving but discontinuous warps, with jump sizes ΔJ = 0.02 and 0.05 at a shared subject-specific location.The jump represents a sudden advance in latent time and violates the model’s continuity assumption.Source: §C.5 · pp. 20–21, Table A3 · p. 20

C.6 Simulation without phase variation

  • With the true warp set to identity, AP-FAE and FAE achieve similar ACC; their MSEs are equal at ρ = 0.2, while AP-FAE has lower MSE at ρ = 0.6.The comparison also includes FAEclust and reports ACC, ATV, and MSE; the cited table caption does not provide additional cell values.Source: §C.6 · p. 21, Table A4 · p. 22

C.7 Scalability

The appendix examines AP-FAE’s behavior across sample sizes and grid resolutions, then provides simulation and real-data visualizations of its fits. The reported scalability results show stable clustering and alignment as these settings vary.

C.7 Scalability

  • Across tested sample sizes, reconstruction error decreases while clustering accuracy and alignment remain relatively stable; grid-resolution changes produce only modest reconstruction and clustering changes.Training time grows approximately linearly with sample size and changes little with grid resolution.Source: §C.7 · p. 22, Table A5 · p. 22, Table A6 · p. 22

C.8 Simulation visualization

  • Figure A1 visualizes a trivariate AP-FAE fit with channels in rows, class colours, and dotted identity lines; Figure A2 shows a fit with B = 0 and ρ = 0.6.Source: Figure A1 · p. 23, Figure A2 · p. 23

C.9 Real-data visualizations

  • Real-data visualizations cover BasicMotions (D = 6), ERing (D = 4), and UWaveGestureLibrary (D = 3), with columns for observations, reconstructions, inverse warps, and aligned curves.Colours indicate classes and dotted lines denote identity.Source: Figure A3 · p. 24, §C.9 · p. 24, Figure A4 · p. 25
  • The remaining real-data fit visualizations cover CBF, Plane, and Symbols, each with D = 1.Across the real-data figures, columns show observed curves, reconstructions, learned inverse warps, and aligned curves; colours indicate classes and dotted lines denote identity.Source: §C.9 · p. 24, Figure A6 · p. 26, Figure A7 · p. 26
Loading 2609.34207v1…